Raphael’s School of Athens: A Theorem in a Painting?

Raphael’s famous painting The School of Athens includes a geometer, presum-ably Euclid himself, demonstrating a construction to his fascinated students. But what theorem are they all studying? This article first introduces the painting, and describes Raphael’s lifelong friendship with the eminent mathematician Paulus of Middelburg. It then presents several conjectured explanations, notably a the-orem about a hexagram (Fichtner), or alternatively that the construction may be architecturally symbolic (Valtieri). The author finally offers his own “null hy-pothesis”: that the scene does not show any actual mathematics, but simply the fascination, excitement, and joy of mathematicians at their work. Raphael’s famous painting The School of Athens shows among the great Greek thinkers at work a geometer, presumably Euclid himself, demonstrat-ing a construction to his students (Figure 1, front right).1 But exactly what theorem is he proving? In this article I describe the best known candidate. I begin with an overview of Raphael’s career and mathematical expertise, then survey the painting as a whole. Next, I focus down on Euclid’s slate–in the painting tilted so sharply that the exact nature of its figure has been a source of controversy–and test whether that figure can in fact arise from an equilateral hexagram (six-pointed star). 1Raphael’s figures in this article are images obtained through Wikimedia, and are in public domain. For readers who would like to see hard print versions, [18] contains well over a hundred beautiful photographs taken by photographers Felice Bono and Pietr